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International Journal of Trend in Scientific Research and Development (IJTSRD)

Volume 4 Issue 4, June 2020 Available Online: www.ijtsrd.com e-ISSN: 2456 – 6470

@ IJTSRD | Unique Paper ID – IJTSRD31618 | Volume – 4 | Issue – 4 | May-June 2020 681

Learning of Advanced Mathematics by Chinese Liberal Arts

Students: A Study of Developing Applied Mathematical Ability

Jingli Liu1, Zhipen Ren2

1Associate Professor, 2Student, 1,2Faculty of Education, Beijing Normal University, Beijing, China

ABSTRACT

According to statistics, more than 90% of Chinese universities have offered

advanced mathematics for liberal arts students as a general course since 2003

to cultivate their scientific literacy and problem-solving ability. This study

aims to provide a feasible approach for liberal arts students to accept the value

of mathematics and apply mathematical methods to academic research and

real-world problems solution. In the Applied Probability and Statistics class

with the attendance of 154 liberal arts freshmen, this study is designed to

discuss and value on the five mathematics application reports as examples.

Then liberal arts students were interviewed on evaluating these 5

mathematics application reports. 28 mathematics application reports were

completed by 154 students in groups and cooperation, which were analyzed

on how mathematics was incorporated into their reports. The qualitative

analysis of the interview results found that the discussion and evaluation of

these 5 mathematics application reports about applying mathematics and

solving social problems can stimulate liberal arts students’ interest in

mathematics and realize the value of mathematics application. Therefore, peer

mathematics application is a way for liberal arts students to realize

mathematics value. Through the classification analysis of the 28 mathematics

application reports completed by the liberal arts students, their three forms of

mathematics application can be summarized as follows: applying the

regularity of mathematical argument instead of the special case of the practice

test in social cognition; using data analysis to monitor the probabilities of

plausible reasoning; developing their applied mathematical ability to solve

daily life problems.

KEYWORDS: Advanced Mathematics, Applied Mathematical Ability, Liberal Arts

Students, Qualitative research method

How to cite this paper: Jingli Liu | Zhipen

Ren "Learning of Advanced Mathematics

by Chinese Liberal Arts Students: A Study

of Developing Applied Mathematical

Ability" Published in

International Journal

of Trend in Scientific

Research and

Development (ijtsrd),

ISSN: 2456-6470,

Volume-4 | Issue-4,

June 2020, pp.1681-

1689, URL:

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1. INTRODUCTION

Chinese liberal arts majors have been offered advanced

mathematics courses for only 40 years. As Yan S.J. (2001)

noted, there were two reasons why liberal arts students

would learn advanced mathematics. The first reason is the

interactions between mathematics and other science fields

have led to the production of new ideas and ways of

thinking. The second reason is that mathematics plays a

crucial role in forming rational thinking. And, since there is a

view that many liberal arts students may lack mathematical

skills and understandings, that then they should learn and

apply mathematics to develop their reasoning ability

(Michal, A., Ruhama E., 2010). In addition, when using

inductive thinking to study various social problems, they

should know that the possibilities afforded by mathematical

reasoning. The application of mathematics knowledge is

extensive, but the application of advanced mathematics for

liberal arts students is waiting to be developed. In the

current advanced mathematics teaching for liberal arts,

students still learn mathematics passively and are lack of

awareness of mathematics application. Therefore, it is

necessary to develop liberal arts students’ applied

mathematics awareness and ability.

As liberal arts students are studying humanities and social

sciences, it can be found that social sciences have different

features from natural sciences, especially the complexity of

the research problems, the causal connection, and the

influence of recognition subject. However, the social

development law revealed by the social sciences has

statistical significance with the unification of probability and

inevitability. At this point, the probabilistic statistical

method can play an important role in social science research.

What is the attitude of mathematics learning for liberal arts

students? It can be seen from the literature that the learning

interest of advanced mathematics for liberal arts students

needs to be improved. According to Gu Pei’s statistics in

2003, more than 90% of Chinese universities have offered

advanced mathematics for liberal arts students. Various

advanced mathematics textbooks have been published that

are suitable for liberal arts students. Sheng L.R.et al. (2005)

proposed that an important means to cultivating students'

innovative ability in universities is to make liberal arts and

mathematics permeate into each other.

IJTSRD31618

International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470

@ IJTSRD | Unique Paper ID – IJTSRD31618 | Volume – 4 | Issue – 4 | May-June 2020 Page 1682

Sheng (2006) tried to offer an additional mathematical

course for liberal arts students to introduce the theory and

method of using mathematics in social science research, e.g.

the feasible region in the linear programming problems, the

fair social distribution, and the probability model in data

analysis, etc. Liberal arts students who have studied this

course were shown to think that such a mathematics course

is more flexible, easier, and practical.

While filling in the literature gap on developing students’

applied mathematics ability in teaching aspect, this paper

aims to make a future contribution in several aspects: (1) to

provide a feasible approach for liberal arts students to

accept the value of mathematics; (2) to help liberal arts

students to solve problems by applying probability and

statistics methods they have learned.

2. Literature Review

The existing literature has mainly focused on the

construction of advanced mathematics courses and teaching

materials, the improvement of teaching methods, and the

increasing of application examples in mathematics textbooks

(Sun Y., 2010), but few focuses on the research of applied

mathematical ability of liberal arts students. Nevertheless, it

is of vital significance to develop their applied mathematics

ability. The advanced mathematics course for liberal arts

should not only help students understand the application of

mathematics but also cultivate their applied mathematics

ability. Here, applied mathematics ability refers to the ability

to analyze and solve practical problems, such as problems in

students’ daily lives and social phenomena, by applying

advanced mathematics knowledge, mathematical logic

thinking and reasoning methods, and the ability to extract

general laws from complicated cases. Advanced mathematics

courses are frequently taught in what is colloquially referred

to as a ‘definition-theorem-proof’ (DTP) format. This format

has its disadvantages, as Davis and Hersh (1981) asserted

that ‘a typical lecture in advanced mathematics consists

entirely of definition, theorem, proof and so on, which can

prevent students from having an intuitive understanding of

the truth about theorems (Hersh, 1993) and discourage

them from using informal ways of understanding

mathematics to produce proofs (e.g., Dreyfus, 1991).

Therefore, the abstraction feature of mathematics may be

difficult for common liberal arts students to understand, and

it is necessary to find suitable ways for liberal arts students

to learn advanced mathematics effectively.

In this study, it is important to develop students’ applied

mathematics ability by writing mathematics application

reports. In the literature, any kind of connection between

mathematics and reality can be regarded as an application of

mathematics (Sloyer, Blum and Huntley, 1995) whereas

applied mathematics is to explain the real world by using the

language of mathematics, answering important questions by

using mathematics and applying mathematics to real life.

Applied mathematics problems may be from the fields of

biology, finance, medicine, business, and so on and require

more complex and longer processes relative to real-life

problems (Pollak, 1976). In this study, the applied

mathematics ability we concern is focused on liberal arts

students in the university so that the application of advanced

mathematics can apply knowledge in daily life or social

science. In these kinds of situations, individuals try to

understand the situations they encounter in their lives,

analyze and interpret them, and reach certain results or

make certain decisions about them (Lubinski, D.,

Humphreys, L. G., 1990).

In order to develop liberal arts students’ applied

mathematics ability, it is important to find out a suitable

approach to teach and learn. Leung, Low, and Sweller (1997)

showed the facts that can help liberal arts students to learn

mathematics with high motivation. They found that students

benefited from verbal explanations of mathematics

problems, at least until they gained a greater facility at

solving these problems. Therefore, it is important for the

liberal arts students in this study to explain mathematical

theorems in their own thinking and language in order to gain

a deeper understanding. Bracha and Kramarski (2006) also

pointed out that the mathematical literacy was closely

related to mathematical problem-solving ability, with the

analysis that PISA usually assesses mathematical literacy in

relation to mathematics curriculum-based knowledge,

mathematical skills, and problem situations in daily life.

With regard to the research method, this study uses a case

analysis method to analyze liberal arts students’ comments

and mathematical application reports, in order to develop

students’ applied mathematics ability. In fact, strong

evidence is available both from mathematics learners’ self-

reporting (e.g., Halmos, 1994) and from research studies

focusing either on interviews with lecturers (e.g., Iannone

and Nardi, 2005) or interviews and observation of practice.

Additionally, Weber (2004) reported a semester-long case

study on how one professor taught real analysis in a

traditional manner, regularly interviewing him about his

teaching practices. Over the past 20 years, the first author in

this paper has taught advanced mathematics to liberal arts

students. To develop the applied mathematical ability of

liberal arts students and find the links between social

science problems and mathematics, this study encouraged

liberal arts students to choose their own research topic,

collect evidence materials and use the knowledge of

advanced mathematics that they have learned in this

semester to solve various specific problems, thereby

developing their applied mathematical ability.

Based on the literature analysis above, this study aims to

provide a feasible approach for liberal arts students to

accept the value of mathematics and apply mathematical

methods to academic research and real-world problems

solution. In the Applied Probability and Statistics class with

the attendance of 154 liberal arts students, this study is

designed to discuss and value on the five mathematics

application reports completed by liberal arts students in the

previous semester. Then liberal arts students were

interviewed on evaluating the topic and method about these

5 mathematics application reports. Finally, mathematics

application reports would be completed by 154 liberal arts

students in groups and cooperation. The research questions

are listed below.

1. What is the feasible approach for liberal arts students to

recognize the value of mathematics?

2. How to qualitatively analyze the role of mathematics in

mathematical application reports completed by liberal

arts students?

To answer these research questions, the case analysis

method was applied to analyze the mathematical application



reports of liberal arts students, and they were interviewed to

elucidate their understanding of these reports.

3. Research Method

In this study, an Applied Probability and Statistics course is

selected. Probability theory and statistics are both applied

mathematical subjects that study and reveal the statistical

laws of uncertain random phenomena. In many fields such as

venture capital and management decision-making or in

people’s daily lives, one will encounter situations requiring

the selection of the right decision in an uncertain situation.

Probability theory provides the most effective theory and

method for solving the uncertainty problem. Statistics is

based on using observational data to analyze and judge

random phenomena and make a reasonable decision.

Probability and statistics are widely used in many fields,

such as natural sciences, management, humanities, and

social sciences. It is because of the wide application of

probability and statistics that we decided that students

should propose research questions independently, apply

mathematics to solve problems based on the research

assumption, and finally write their mathematics application

reports.

The sample for this study was selected from a general

education course at a university in China. At this university

(and almost all universities in China), liberal arts students

must study advanced mathematics in general education. The

first-year general education compulsory course Applied

Probability and Statistics was selected in this study. For the

post-class mathematics homework, the liberal arts students

were required to complete mathematics problem-solving

assignments after each class. In addition, they were required

to complete mathematical application reports by group

discussion and cooperation throughout the semester.

This research is based on the previous work lasting for over

10 years, the purpose of which is trying to encourage liberal

arts students to develop their mathematics application

reports. During this period, 205 math application reports

developed by 1123 liberal arts students have been collected

(Authors, 2018). Before encouraging liberal arts students to

develop mathematics application reports, examples of

previous mathematics application reports among different

subjects in the early stage were presented to make sure that

liberal arts students can fully discuss the content and

characteristics of these example reports. On the basis of

previous experience, the high quality of mathematics

application reports written by liberal arts students is

guaranteed. Therefore, the procedure of this study is as Fig 1.

Figure 1 Research path of this study

This study was conducted in the Applied Probability and Statistics course in one semester spanning February to July 2019. The

subjects were 154 freshmen who majored in pedagogy, education management, preschool education, and special education.

Among them, 80% were female, and 20% were male; 76% of them had studied algebra, trigonometric function, preliminary

solid geometry, plane analytic geometry, calculus, statistical cases, reasoning and proof, and expansion of the number system in

high school while others had studied extra mathematics topics such as vector and solid geometry, permutation and

combination, and probability. In Applied Probability and Statistics course, students need to study probabilistic and statistical

knowledge, such as classical probability, geometry approach, conditional probability, total probability Formula, Bayes Formula,

Bernoulli probability, binomial distribution, normal distribution, mathematical expectation, variance, central limit theorem;

descriptive statistics, correlation analysis, sampling, point estimation, interval estimation, etc. (Authors, 2016). At the

beginning of the new semester in this study, five of these mathematical application reports were selected and presented to the

154 subjects in class in order to give them a demonstration and inspiration. Regarding the fields covered in the five reports,

two belonged to science fields, two belonged to social sciences, and one pertained to students’ daily lives. The 154 liberal arts

students were divided into 28 groups to discuss the features of the 5 presentation reports. Each group had a group leader

selected by the members, and they would be taken an interview to make comments on the 5 example reports. Then, each group

discussed its own research topic and implemented the process of writing the mathematical application report. The groups were

required to submit their mathematical application reports as group assignments within one semester. During the semester, the

organization of data collection could be divided into two cycles, as described below. The first author was the teacher of this

course and had a history of involvement in such endeavors dating back 20 years.



3.1. Five Reports as Examples of Mathematical Application

At the beginning of the semester, five presentations were selected from the previous reports based on the creation of topic

selection, the proper application of mathematical knowledge, the logic of argument and the essential role of mathematics in

unexpected conclusions over the past 10 years. These five reports were recorded as R1, R2, R3, R4, and R5. Their titles are as

follows.

R1: Which one is more difficult, receiving an offer from Tsinghua University or winning the first-class lottery?

It is the dream for most Chinese high school students to pass the entrance examination of Tsinghua University, and it is also

human nature to desire to win the lottery to become wealthy. Six girls as a group in this study investigated the difficulty of

receiving an offer from Tsinghua University versus winning the first-class lottery. This group collected the data about the

winning probability of this lottery and calculated that the first-class probability is 1/17,720,000. As a comparison, this group

collected the mean and variance data of passing the entrance examination of Tsinghua University and found out that the

probability of a high-school student getting the offer from Tsinghua University is about 1.5%. Apparently, the probability of

getting an offer from Tsinghua University is higher than the probability of winning the first-class lottery.

R2: What is the use of the Bayesian formula?

The Bayesian formula will be taught in Applied Probability and Statistics. Three example questions in the textbook use the

Bayesian formula to solve applied problems. However, a group in this study has solved three example questions without using

the Bayesian formula but using "addition principle" and "multiplication principle" in an easier way. The group leader believes

that the Bayesian formula could be too complicated to use, and some problems could also be solved without it.

R3: It is hard for the “3-high” female to get married.

There is a specialized social problem in some big cities in China where the "3-high" female, which means women with high IQ,

high education, and high salary, are less likely to get married. A group in this study collected the birth ratio data of men and

women in China. This group applied the normal distribution knowledge learned from the Applied Probability and Statistics

course and calculated that according to the principle of 3-σ principle of normal distribution, people with normal IQ accounts for

account for 34.1%, people with high IQ accounts for 13.6%, people with higher IQ accounts for 2.14%, and people with super

higher IQ accounts for 0.16%. The “3-high” female are already in the 13.6% of the total people and they would like to marry

men with higher IQ than themselves in China, which can only be found in the 2.14% of the total male people. The gender ratio

of China's population at birth comes from the data of the China Statistical Yearbook. The average IQ of people is quoted from

professional literature data, while high IQ is the value defined by liberal arts students themselves that are higher than the

average IQ.

R4: Star-chasing students’ winning-ticket process

This report has a background about a famous Chinese singer, Yi, who was 18 years old in 2018, the same age as freshmen in

this study. A group of 6 fans completed this report to calculate the probability of getting a concert ticket of Yi in different ways.

They collected related data and calculated the probability of getting the ticket. The number of tickets in a different way is: 1,200

from the performance company, 100 from microblogs of Yi, 300 from Taobao.com, and 200 from Tmall.com. Then data showed

that the number of buyers is: 30,158 from the performance company, 910,000 from microblogs of Yi, 161,000 from

Taobao.com, and 340,000 from Tmall.com. Assuming the probability of a fan buying this ticket is 0.3 at the performance

company, 0.1 at microblogs of Yi, 0.3 at Taobao.com, and 0.3 at Tmall.com, the probability of this fan buying this ticket

successfully is 0.0127. It is noted that the data are collected from the performance company website, microblog of Yi,

Taobao.com and Tmall.com by this group.

R5: Reasonable blood test issues

The mathematical knowledge in R5 is the mathematical expectation. It draws lessons from a large number of conscriptions in

the United States during World War II. To check whether the candidates suffer from certain diseases, group blood tests are

used. If the blood of a group (k candidates) is qualified, only one test will be performed. If it is not qualified, a total of k+1 tests

will be conducted, one for each of the candidates. This is a grouping inspection method that both ensures the elimination of

unqualified candidates and reduces the workload of the method. The liberal arts students in this study applied this method to

other problems, such as detecting whether water from different sources is polluted. According to the method mentioned above,

it is possible to draw reliable conclusions without adding experimental equipment.

The five cases were divided into scientific situations (R2, R5), social science situations (R1, R3), and students’ daily life

situations (R4).

3.2. Cycle 1: The Purpose of Presenting Reports as Examples to Participants

In the first class, 154 liberal arts students received five examples of mathematical applications and were divided into 28 groups

to discuss the five reports. The first cycle of data collection consisted of lecture observations and the email collection of

subjects’ comments about the 5 reports presented in class. Table 1 summarizes the data collected in this cycle.



Table 1 Breakdown of data collected for cycle 1

Step Data collected Purpose

1

5 of the previous mathematical application

reports were selected and presented to the

154 subjects in the class

To evaluate 5 examples of mathematical

application reports and find out the connection

with the probability and statistics knowledge

2 The groups were required to discuss the 5

presented reports

To analyze the role of mathematics in the 5

mathematical application reports

3 28 groups submitted their comments by

email, and some of them were interviewed

Liberal arts students are expected to write their

own mathematical application reports

It is necessary to note that cycle 1 of the data collection included the discussion of the five typical reports and subsequent

interviews with students and that the reports were NOT written that semester but rather in the previous semester. At the

beginning of the semester, the liberal arts students had not learned the mathematical knowledge contained in the five reports,

but they can see the arguments and demonstrations of the mathematical formulas and calculations in the reports and also

evaluate the impression that humanities and social sciences can be integrated into mathematics.

3.3. Cycle 2: Design Process of Mathematical Application Reports

The first cycle of data collection consisted of the mathematical application reports created by the 154 liberal arts students

themselves and the interview records of some subjects. During the semester, the liberal arts students cooperated within groups

to discuss their ideas, design their reports, and finally submit their reports to their math teacher. Most of them presented their

reports in front of the whole class. It is necessary to note that mathematics knowledge in these reports mostly covered the first

half of the semester because most of the subjects submitted and presented their mathematical application reports after the

middle of the semester. Table 2 summarizes the data collected in this cycle.

Table 2 Breakdown of data collected for cycle 2

Step Data collected Purpose

1 Topics of the 28 mathematics application reports To classify the topics of 28 mathematics application reports

2 The applied mathematics cases from 28

mathematics application reports

To analyze the role of mathematics in the 28 mathematical

application reports

3 Reflections on the process of report creation To understand and master mathematical method in solving

problems

In addition, for this cycle, the mathematical application reports were text recorded, and interviews were audio-recorded by

email with verbatim transcription.

This section is divided into two parts: 154 liberal arts students were divided into 28 groups to evaluate five examples of

mathematical application reports, and 28 groups wrote new math application reports during the semester.

4. Findings

4.1. Evaluation of the Five Reports by 28 Student Representatives

In this study, 5 previous mathematical application reports were presented to 154 students in the first lesson of advanced

mathematics for liberal arts courses as an example model. The students were divided into 28 groups to discuss these reports in

groups, and they then submitted their comments. Finally, each group elected a representative whose comments were received

by e-mail, and they were recorded as S1, S2, etc. NVivo 10® software was used to edit the 28 written documents and classify

them into nodes. Each node corresponds to a topic from a category of co-participants.

The research path of grounded theory was used in this study to import open interview problems and report texts into NVivo

10® software for sorting and coding. This study has repeatedly and continuously compared, analyzed, and generalized the

mathematical application reports developed by liberal arts students as well as the interviews based on students’ self-evaluation

about their mathematical application reports. The theory has finally constructed after gradually conceptualizing and classifying

them from low level to high level. The coding of this study can be divided into three levels: the first level open coding, the

second level relational coding, and the third level core coding.

First level open coding: interview and report texts were decomposed, compared, conceptualized, and categorized to keep the

coding open and close to the data. The two researchers first encoded all the items independently and then tested the

consistency of the codes. The results showed that the consistency ratio reached 92%, and the coding methods and rules were

clear and definite in general. For the inconsistencies, the coder could be discussed again until an agreement was reached. In the

process of first level open coding, researchers have tried to keep the key elements of the text, but at the same time, some

colloquial expression was removed, so that a total of 28 first-level codes were formed.

Second level relational coding: based on the first level open coding, genera, and dimensions could be formed to develop and test

the relationship between genera and forms related categories. Thus, 8 relational codes were formed.

Third level core coding: after a systematic analysis of all the discovered conceptual genera, the core genera with high generality

and integrity were selected, thus forming 4 core codes., or 4 themes, as shown in Table 3.



Table 3 Student’s comments about 5 presented reports

Theme Comments

Theme 1: Liberal arts students’ understanding of

applied mathematics

(1) The results applied with mathematics are more

objective compared with human’s subjective wishes

(2) Mathematical knowledge can be meaningful only if

applied in the real world.

Theme 2: The connections between mathematics

and topics from mathematical application reports

(1) Social cognitive problems.

(2) Social phenomena.

Theme 3: The role of mathematics in the

mathematical application reports

(1) Mathematics can provide objective judgment for

social cognition.

(2) Mathematical methods can be regarded as tools for

explaining social phenomena.

Theme 4: The challenge of mathematics

knowledge from liberal arts students

(1) It is unfeasible to blindly follow the complex

mathematical formula.

(2) Learn to pass on some opinions through deduction,

explanation, and recitation of mathematical formulas

in real situations.

4.1.1. Theme 1: Liberal Arts Students' Understanding of Applied Mathematics

S10: In report R4, celebrity-chasing girls quoted online sales statistics about celebrity birthday tickets and then used

conditional probability and full probability to calculate that the probability of purchasing tickets successfully is very small. The

conclusion is regrettable but acceptable given the data support behind it. Some liberal arts students realized that the results

applied in mathematics are more objective compared with our subjective wishes.

S11: Without a deep understanding of the topic itself, the solution to the problem will be too rigid; without the ability to grasp

the key points, the problem solving will become complicated. In fact, every person’s brain stores a considerable number of

problem-solving methods, but these methods are often disorganized. When you need to solve a specific problem, due to the

confusion of these methods, we prioritize the most familiar methods. Although mathematics method is not a familiar method

for liberal arts students themselves, they can see from the 5 reports that mathematics is still available, and is only useful in the

process of solving problems.

Therefore, liberal arts students agreed that mathematical knowledge can be meaningful only if applied in the real world.

4.1.2. Theme 2: The Connections Between Mathematics and Topics from Mathematical Application Reports

S5: The report took admission to Tsinghua University and 5 million yuan to be the desires of our youth. These goals attract a

very high degree of attention such that they easily to resonate. College students have tried hard to get admitted to prestigious

schools and know the difficulties. Such topics give us a sense of substitution. This case is different from the general discussion

in society. It applies the probability calculation as the entry point, uses the data to express a viewpoint, and inspires deep

thinking with its conclusion. This case shows the connection between social cognitive problems and mathematics.

S9: The reason why it is impressive that 3-high women have difficulty marrying is that this case reflects the application of

statistical probability problems to a general phenomenon of society, indicating that mathematics is not just for calculus or

numerical formulas but also relevant to real-life for solving problems. This case shows the connection between social

phenomena and mathematics.

4.1.3. Theme 3: The Role of Mathematics in the Mathematical Application Reports

S15: The expressions of social hotspots are diverse, and when they are analyzed by statistics, their essence is mostly the same.

Some time ago, a commercial company held an activity about a large-scale lottery “Koi”, which became a hot topic. Through the

case analysis of small-probability events, people can have a more rational understanding of such activities, and they will not

blindly believe that small-probability events come true easily. Therefore, mathematics can provide objective judgment for

social cognition.

S14: Report R4 is closely related to the students, so it is easy to attract students’ attention. The analysis process of report R4 is

as follows: asking questions → collecting data → calculating probability → relying on the results to answer the questions raised

before→ rethinking the application of conclusions. Most of the data are from an ofUicial source, which is more authoritative, and

the probability calculation method is appropriate. The results are not only quite accurate and convincing but also valuable.

Therefore, mathematical methods can be regarded as tools for explaining social phenomena.

4.1.4. Theme 4: The challenge of mathematics knowledge from liberal arts students

S10: Thinking stereotypes have a great influence on people's problem-solving. Sometimes they are a good thing to help us solve

common problems as soon as possible, but sometimes they limit the divergence of people's thinking, which makes it difficult for

us to solve problems creatively. When R2 solves the specific situation, the author can solve the problem by choosing suitable

math methods. This is what we students should learn: the mathematical application should not blindly follow knowledge from

textbooks but should create our own innovative thinking.



S24: The case What is the use of the Bayesian formula can let us break through the shackles of thinking and dare to question

authority, which is a kind of courage and praiseworthy; this case is good at thinking rather than blindly excluding with its own

theoretical methods, solution system, and valuable innovative thinking. Many students may have glimpsed doubts about the

use of Bayesian formula in their minds, but most of them do not have enough confidence to compete with authority, or they do

not think deeply and try to solve problems in their own way. Innovation is a kind of knowledge transfer and application.

Innovation on the basis of a deep understanding of existing mathematical knowledge is a new way to explore and solve

problems. This author’s example realized me that liberal arts students should not learn mathematical theories in an abstract

way, but should transfer some opinions through deduction, explanation and recitation of mathematical knowledge in real

situations.

4.2. The Evaluation of 28 Mathematical Application Reports in This Semester

In the Applied Probability and Statistics course in the 2018 semester, 154 liberal arts students were required to divide into 28

cooperative groups and complete a mathematical application report within one semester. Among 28 mathematical application

reports, there were 3 scientific situations (7%), 2 social situations (11%) and 23 life situations (82%), as shown in Table 4.

Table 4 Question Situations in 28 Completed Mathematics Application Reports

Question Situations

(number, percent) Example Topics The Role of Mathematics

Scientific Situations

(3, 11%)

Probability interpretation of

Murphy's law

Applying the regularity of mathematical argument instead

of the special case of the practice test in social cognition

Social situations

(2, 7%)

Probability thinking from a

physical education examination

Using data analysis to monitor the probabilities of

plausible reasoning

Daily life situations

(23, 82%)

The choice of lunch using

probability

Developing their applied mathematical ability to solve

daily life problems

In the first question situations, scientific situations, liberal

arts students applied the regularity of mathematical

argument instead of the special case of the practice test in

social cognition. For instance, Edward A. Murphy's Law

(1949) shows that if bad things have the possibility to

happen, it will happen and cause the most possible loss, no

matter how small the possibility is. Most liberal arts students

agree with Murphy's law based on real cases in their lives.

This group assumed that the completion of some complex

event included a total of m steps and n results. They

calculated out that the probability of a good outcome is much

lower than that of the bad outcome by applying the addition

principle and the multiplication principle. They concluded

with logical inference if we want to achieve something

successfully, we must eliminate all the bad possibilities and

avoid all the pits perfectly. We can see that this group can

apply probability calculation as proof of their social

cognition and logical inference.

The second question situations are social situations. Taking

the reports of the probability thinking from a physical

education examination as an example, liberal arts students

believed that the scoring standard of college basketball

course, which is evaluated with the number of goals shot in

10 times, is unreasonable. They questioned that the final

score could not represent students’ usual percentage of

pitches? They calculated that if someone's hitting probability

was 0.6 in normal times, then the probability of 6 pitches

was 0.25082 by using the Bernoulli formula, which is smaller

than we had expected. Therefore, reasonable reasoning also

needs data analysis and monitoring. Liberal arts students

made a conclusion that reasonable reasoning is necessary for

problems solution, but there may be deviations due to

various unpredictable reasons. Therefore, data analysis can

monitor the probability of reasonable reasoning.

The third question situations, daily life situations, are to use

mathematics as a tool to solve problems in students’ daily

life. For example, the case the choice of lunch has used

probability knowledge to optimize the choice of lunch in

university life. Through data collection, they found out that

students’ rest time at noon is limited. In order to ensure

punctual attendance in the afternoon class, this group would

like to know about lunchtime in the different canteen with a

95% confidence interval. The data of the lunch period of 20

students in each canteen was collected, and the range of the

average lunch period was estimated by using the interval

estimation method. Specifically, after data collection, they

calculated the mean value and standard deviation, and then

concluded that the 0.95 confidence level of lunch period in

the 5th canteen is (29.12min, 44.48min) by using the

confidence interval of the overall mean μ; the 0.95

confidence level of lunch period in the 4th canteen is

(39.42min, 49.59min); and so on. If only considering the

time factor, it is more appropriate to choose the 5th canteen

for lunch in order to guarantee punctual attendance in the

afternoon class.

The qualitative analysis of the interview results found that

the discussion and evaluation of these 5 mathematics

application reports about applying mathematics and solving

social problems. When discussing the mathematical

application reports on solving some social and daily-life

problems by applying mathematical knowledge, liberal arts

students’ interest in mathematics and realize the value of

mathematics application can be simulated. Therefore, peer

mathematics application is a way for liberal arts students to

realize mathematics value. Through the classification

analysis of the 28 mathematics application reports

completed by the liberal arts students, their three forms of

mathematics application can be summarized as follows:

applying the regularity of mathematical argument instead of

the special case of the practice test in social cognition; using

data analysis to monitor the probabilities of plausible

reasoning; developing their applied mathematical ability to

solve daily life problems.

5. Discussions

The 28 mathematical application reports were the result of

interaction between liberal arts students and social



situations. Laplace (1829), a French mathematician, always

emphasized the practicability of probability theory and

mentioned more than once that the delicate calculation in

probability theory can be applied to the most important

problems in daily life. Among practical sciences, there are

uncertainties that we cannot understand and describe, but

probability theory is just a mathematical tool to make up for

our lack of knowledge. Therefore, Applied Probability and

Statistics as a general university course has both the

scientific nature of mathematical knowledge and the

sociality of content situations, and it can satisfy people's

curiosity about predicting the future. 82% of the 28

mathematical application reports in the 2019-semester

Applied Probability and Statistics course focused on daily

situations, and the students see the wisdom of the

probability and statistics with small scenarios so that people

can understand the complex mathematics knowledge while

paying attention to the reality of life.

Of relevance to the present study, it is noteworthy that the

28 mathematics application reports developed by the liberal

arts students in this study took into account the

characteristics of both mathematics and literature because

their research problems were specific and practical, their

research ideas had mathematical rational thinking, and their

research conclusions were based on evidence and easily

relatable to other students.

It can be seen from the mathematical application reports and

interviews of liberal arts students that they don't value math

as a problem-solving tool, but value mathematics

application. Liberal arts students have opened up a new field

of applied mathematics, and they have applied mathematics

to social cognition, social phenomena, and other issues.

These reports may open up a new direction of mathematical

application, and applied mathematics can also enrich the

social research methods of liberal arts students.

6. Research Limitations and Future Research

Directions

6.1. Research Limitations

There are three limitations to the intent and the scope of the

study. Firstly, regarding a small number of participants, the

participants are only limited to liberal arts students who

majored in pedagogy, education management, preschool

education, special education and other limited majors.

Whether the research results are suitable for other liberal

arts students or not needs further verification. Secondly,

liberal arts students have completed their mathematics

application reports by group cooperation and each

member's contribution to their report is required to be

noted. However, how the mathematical application reports

are designed, how the students actively interact and develop

with each other and between individuals and groups, remain

to be further investigated.

6.2. Future Research Directions

We suggest three broad ideas in future research. The first is

to collect more cases in a variety of liberal arts majority and

universities in different countries to help distinguish

between local cultures and global cultures about the

mathematical application. It is also important to find out the

difference in the mathematical application process between

different majority and universities. The second is to explore

the application in other mathematical fields, such as the

application of statistics for liberal arts students. Finally, this

is a question that is still open yet important to investigate

liberal arts students’ learning styles and mental frames

interact when applying mathematics in groups.

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